Article ID: MTJPAM-D-20-00006

Title: The Faber Polynomial Expansion Method for a Subclass of Analytic and Bi-Univalent Functions Associated with the Janowski Functions


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00006; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 17-24

Document Type: Research Paper

Author(s): Nazar Khan a , Qazi Zahoor Ahmad b, Shahid Khan c , Bilal Khan d

aDepartment of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan

bDepartment of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan

cDepartment of Mathematics Riphah International University Islamabad, Pakistan

dSchool of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, People’s Republic of China

Received: 27 March 2020, Accepted: 1 December 2020, Published: 25 April 2021.

Corresponding Author: Nazar Khan (Email address: nazarmaths@gmail.com)

Full Text: PDF

Abstract

In this present investigation, we introduce a new subclass of analytic and bi-univalent functions associated with Janowski functions. Using the Faber polynomial expansions, we determine a general coefficients bounds |an|, n ≥ 3 for this newly defined class. Relevant connections of the results presented in this paper with those in a number of other related works on this subject are also pointed out.

Keywords: Analytic functions, Univalent functions, Bi-univalent functions, Faber polynomial expansion

References:
  1. H. Airault, Symmetric sums associated to the factorizations of Grunsky coefficients, in Conference, Groups and Symmetries. Montreal, Canada, 2007.
  2. H. Airault, Remarks on Faber polynomials, Int. Math. Forum. 3 (9), 449-456, 2008.
  3. H. Airault and H. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math. 130 (3), 179-222, 2006.
  4. Ş. Altnkaya and S. Yalçin, Faber polynomial coefficient bounds for a subclass of bi-univalent functions, C. R. Acad. Sci. Paris, Ser. I 353, 1075-1080, 2015.
  5. D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, in: S.M. Mazhar, A. Hamoui, N.S. Faour (Eds.), Mathematical Analysis and its Applications, Kuwait, 1985, in: KFAS Proceedings Series, vol. 3, Pergamon Press, Elsevier Science Limited, Oxford, 1988, pp. 53-60; see also Studia Univ. Babe3-Bolyai Math. 31 (2), 70-77, 1986.
  6. S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions, C. R. Acad. Sci. Paris, Ser. I 352, 479-484, 2014.
  7. P. L. Duren, Univalent Functions, Grundehren der Math. Wiss., Vol. 259, Springer-Verlag, New York, 1983.
  8. G. Faber, Uber polynomische Entwickelungen, Math. Ann. 57 (3), 1569-1573, 1903.
  9. S. G. Hamidi and J. M. Jahangiri, Faber polynomial coefficients of bi-subordinate functions, Comptes Rendus Mathematique. 354, 365-370, 2016.
  10. W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28, 297-326, 1973.
  11. S. Mahmood, Q. Z. Ahmad, H. M. Srivastava, N. Khan, B. Khan and M. Tahir, A certain subclass of meromorphically q-starlike functions associated with the Janowski functions, J. Inequal. Appl. 2019 (88), 1-11, 2019.
  12. M. Sabil, Q. Z. Ahmad, B. Khan M. Tahir and N. Khan, Generalisation of certain subclasses of analytic and univalent functions, Maejo Internat. J. Sci. Technol. 13 (01), 1-9, 2019.
  13. A. C. Schiffer and D. C. Spencer, The coefficient of Schlicht functions, Duke Math. J. 10, 611-635, 1943.
  14. M. Schiffer, A method of variation within the family of simple functions, Proc. Lond. Math. Soc. 44 (2), 432-449, 1938.
  15. H. M. Srivastava, Q. Z. Ahmad, N. Khan, N. Khan and B. Khan, Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain, Mathematics 7 (2), 181, 1-15, 2019.
  16. H. M. Srivastava, Ş. Altinkaya and S. Yalçin, Certain subclasses of bi-univalent functions associated with the Horadam polynomials, Iran. J. Sci. Technol. Trans. A: Sci. 43, 1873-1879, 2019.
  17. H. M. Srivastava, S. Bulut, M. Caglar and N. Yagmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat 27, 831-842, 2013.
  18. H. M. Srivastava, S. Khan, Q. Z. Ahmad, N. Khan and S. Hussain, The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain q-integral operator, Stud. Univ. Babeş-Bolyai Math. 63, 419-436, 2018.
  19. H. M. Srivastava, B. Khan, N. Khan and Q. Z. Ahmad, Coefficient inequalities for q-starlike functions associated with the Janowski functions, Hokkaido Math. J. 48, 407-425, 2019.
  20. H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23, 1188-1192, 2010.
  21. H. M. Srivastava, A. Motamednezhad and E. A. Adegan, Faber polynomial coefficient estimates for bi-univalent functions defined by using differential subordination and a certain fractional derivative operator, Mathematics 8 (2), 172, 1-12, 2020.
  22. H. M. Srivastava, S. Sümer Eker and R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat 29 (8), 1839-1845, 2015.
  23. H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad and N. Khan, Some general classes of q-starlike functions associated with the Janowski functions, Symmetry 11 (2), 292, 1-14, 2019.
  24. H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad and N. Khan, Some general families of q-starlike functions associated with the Janowski functions, Filomat 33 (9), 2613-2626, 2019.
  25. H. M. Srivastava and A. K. Wanas, Initial Maclaurin coefficient bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination, Kyungpook Math. J. 59, 493-503, 2019.
  26. T. S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981.
  27. M. Tahir, N. Khan, Q. Z. Ahmad, B. Khan and G. Mehtab, Coefficient estimates for some subclasses of analytic and bi-univalent functions associated with conic domain, Sahand Communications in Math. Anal. (SCMA) 16 (1), 69-81, 2019.
  28. Q.-H. Xu, H.-G. Xiao and H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218, 11461-11465, 2012.